Section 10.9 Polar Equations of Conics 793
What you should learn• Define conics in terms of
eccentricity.
• Write and graph equations of conics in polar form.
• Use equations of conics inpolar form to model real-lifeproblems.
Why you should learn itThe orbits of planets and satellites can be modeled withpolar equations. For instance, inExercise 58 on page 798, a polarequation is used to model theorbit of a satellite.
Polar Equations of Conics
Digital Image © 1996 Corbis; Original image courtesy of NASA/Corbis
10.9
Alternative Definition of ConicIn Sections 10.3 and 10.4, you learned that the rectangular equations of ellipsesand hyperbolas take simple forms when the origin lies at their centers. As ithappens, there are many important applications of conics in which it is moreconvenient to use one of the foci as the origin. In this section, you will learn thatpolar equations of conics take simple forms if one of the foci lies at the pole.
To begin, consider the following alternative definition of conic that uses theconcept of eccentricity.
In Figure 10.77, note that for each type of conic, the focus is at the pole.
Polar Equations of ConicsThe benefit of locating a focus of a conic at the pole is that the equation of theconic takes on a simpler form. For a proof of the polar equations of conics, seeProofs in Mathematics on page 808.
Alternative Definition of ConicThe locus of a point in the plane that moves so that its distance from a fixedpoint (focus) is in a constant ratio to its distance from a fixed line (directrix)is a conic. The constant ratio is the eccentricity of the conic and is denotedby Moreover, the conic is an ellipse if a parabola if and ahyperbola if (See Figure 10.77.)e > 1.
e � 1,e < 1,e.
Polar Equations of ConicsThe graph of a polar equation of the form
1. or 2.
is a conic, where is the eccentricity and is the distance betweenthe focus (pole) and the directrix.
�p�e > 0
r �ep
1 ± e sin �r �
ep
1 ± e cos �
2
F = (0, 0)
QP
0
π
Directrix
Parabola:PFPQ
� 1
e � 1
2
F = (0, 0)
Q P
0
Directrix
π
Ellipse:
FIGURE 10.77
PFPQ
< 1
0 < e < 1
2
F = (0, 0)
Q
Q ′P ′
P
0
Directrix
π
HyperbolaPFPQ
�P�F
P�Q�> 1
e > 1
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Equations of the form
Vertical directrix
correspond to conics with a vertical directrix and symmetry with respect to thepolar axis. Equations of the form
Horizontal directrix
correspond to conics with a horizontal directrix and symmetry with respect to theline Moreover, the converse is also true—that is, any conic with a focusat the pole and having a horizontal or vertical directrix can be represented by oneof the given equations.
� � ��2.
r �ep
1 ± e sin �� g�sin ��
r �ep
1 ± e cos �� g�cos ��
For the ellipse in Figure 10.78, the major axis is horizontal and the verticeslie at and So, the length of the axis is To find thelength of the axis, you can use the equations and to conclude that
Ellipse
Because you have which implies thatSo, the length of the minor axis is A similar
analysis for hyperbolas yields
Hyperbola � a2�e2 � 1�. � �ea�2 � a2
b2 � c2 � a2
2b � 6�5.b � �45 � 3�5.b2 � 92�1 � �2
3�2� � 45,e �23,
� a2�1 � e2�. � a2 � �ea�2
b2 � a2 � c2
b2 � a2 � c2e � c�aminor2a � 18.major�3, ��.�15, 0�
794 Chapter 10 Topics in Analytic Geometry
Identifying a Conic from Its Equation
Identify the type of conic represented by the equation r �15
3 � 2 cos �.
Example 1
Algebraic SolutionTo identify the type of conic, rewrite the equation inthe form
Write original equation.
Because you can conclude that the graphis an ellipse.
Now try Exercise 11.
e �23 < 1,
�5
1 � �2�3� cos �
r �15
3 � 2 cos �
r � �ep���1 ± e cos ��.
Graphical SolutionYou can start sketching the graph by plotting points from to Because the equation is of the form thegraph of is symmetric with respect to the polar axis. So, youcan complete the sketch, as shown in Figure 10.78. From this,you can conclude that the graph is an ellipse.
rr � g�cos ��,� � �.
� � 0
Divide numerator anddenominator by 3.
0
2
π(3, ) (15, 0)
3 6 18 219 12
θr = 153 2 cos−
π
FIGURE 10.78
Additional ExampleIdentify the conic and sketch its graph.
Solution
Parabola
0
2π
1 2 4 5
r �4
2 � 2 sin �
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Sketching a Conic from Its Polar Equation
Identify the conic and sketch its graph.
SolutionDividing the numerator and denominator by 3, you have
Because the graph is a hyperbola. The transverse axis of the hyperbolalies on the line and the vertices occur at and Because the length of the transverse axis is 12, you can see that To find
write
So, Finally, you can use and to determine that the asymptotes of thehyperbola are The graph is shown in Figure 10.79.
Now try Exercise 19.
In the next example, you are asked to find a polar equation of a specifiedconic. To do this, let be the distance between the pole and the directrix.
1. Horizontal directrix above the pole:
2. Horizontal directrix below the pole:
3. Vertical directrix to the right of the pole:
4. Vertical directrix to the left of the pole:
Finding the Polar Equation of a Conic
Find the polar equation of the parabola whose focus is the pole and whosedirectrix is the line
SolutionFrom Figure 10.80, you can see that the directrix is horizontal and above thepole, so you can choose an equation of the form
Moreover, because the eccentricity of a parabola is and the distancebetween the pole and the directrix is you have the equation
Now try Exercise 33.
r �3
1 � sin �.
p � 3,e � 1
r �ep
1 � e sin �.
y � 3.
r �ep
1 � e cos �
r �ep
1 � e cos �
r �ep
1 � e sin �
r �ep
1 � e sin �
p
y � 10 ± 34 x.
bab � 8.
� 64.� 62�5
32
� 1�b2 � a2�e2 � 1�
b,a � 6.��16, 3��2�.�4, ��2�� � ��2,
e �53 > 1,
r �32�3
1 � �5�3� sin �.
r �32
3 � 5 sin �
Section 10.9 Polar Equations of Conics 795
Example 2
Example 3
0
2
π
π
θr = 323 + 5 sin
4 8
4,
−16,
2
2
( )
( )3
π
FIGURE 10.79
0
2
θr = 31 + sin
(0, 0) 1 3 42
Directrix:y = 3
π
FIGURE 10.80
Use a graphing utility set in polar mode to verify the four orientations shown at the right.Remember that e must bepositive, but p can be positive ornegative.
Techno logy
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ApplicationsKepler’s Laws (listed below), named after the German astronomer JohannesKepler (1571–1630), can be used to describe the orbits of the planets about the sun.
1. Each planet moves in an elliptical orbit with the sun at one focus.
2. A ray from the sun to the planet sweeps out equal areas of the ellipse in equaltimes.
3. The square of the period (the time it takes for a planet to orbit the sun) isproportional to the cube of the mean distance between the planet and the sun.
Although Kepler simply stated these laws on the basis of observation, they werelater validated by Isaac Newton (1642–1727). In fact, Newton was able to showthat each law can be deduced from a set of universal laws of motion andgravitation that govern the movement of all heavenly bodies, including cometsand satellites. This is illustrated in the next example, which involves the cometnamed after the English mathematician and physicist Edmund Halley(1656–1742).
If you use Earth as a reference with a period of 1 year and a distance of 1astronomical unit (an is defined as the mean distance betweenEarth and the sun, or about 93 million miles), the proportionality constant inKepler’s third law is 1. For example, because Mars has a mean distance to the sunof astronomical units, its period is given by So, the periodof Mars is years.
Halley’s Comet
Halley’s comet has an elliptical orbit with an eccentricity of Thelength of the major axis of the orbit is approximately 35.88 astronomical units.Find a polar equation for the orbit. How close does Halley’s comet come to thesun?
SolutionUsing a vertical axis, as shown in Figure 10.81, choose an equation of the form
Because the vertices of the ellipse occur when and you can determine the length of the major axis to be the sum ofthe -values of the vertices. That is,
So, and Using this value of in theequation, you have
where is measured in astronomical units. To find the closest point to the sun(the focus), substitute in this equation to obtain
Now try Exercise 57.
� 55,000,000 miles.� 0.59 astronomical unit r �1.164
1 � 0.967 sin���2�
� � ��2r
r �1.164
1 � 0.967 sin �
epep � �0.967��1.204� � 1.164.p � 1.204
2a �0.967p
1 � 0.967�
0.967p
1 � 0.967� 29.79p � 35.88.
r� � 3��2,
� � ��2r � ep��1 � e sin ��.
e � 0.967.
P � 1.88d3 � P2.Pd � 1.524
unitastronomical
796 Chapter 10 Topics in Analytic Geometry
Example 4
Activities
1. Identify the conic and sketch itsgraph.
Answer: Ellipse
2. Find a polar equation of the parabolawith focus at the pole and directrix
Answer: r �2
1 � sin �
y � 2.
0
2π
1 2 3 5
r �4
3 � 2 cos �
π 0
2
π2
3
Halley’scomet
Sun
Earth
π
FIGURE 10.81
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Section 10.9 Polar Equations of Conics 797
In Exercises 1–4, write the polar equation of the conic forand Identify the conic for each
equation. Verify your answers with a graphing utility.
1. 2.
3. 4.
In Exercises 5–10, match the polar equation with its graph.[The graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a) (b)
(c) (d)
(e) (f)
5. 6.
7. 8.
9. 10.
In Exercises 11–24, identify the conic and sketch its graph.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
In Exercises 25–28, use a graphing utility to graph the polarequation. Identify the graph.
25. 26.
27. 28. r �4
1 � 2 cos �r �
3
�4 � 2 cos �
r ��5
2 � 4 sin �r �
�1
1 � sin �
r �2
2 � 3 sin �r �
4
2 � cos �
r �3
2 � 6 sin �r �
3
2 � 6 cos �
r �5
�1 � 2 cos �r �
3
2 � 4 sin �
r �9
3 � 2 cos �r �
6
2 � sin �
r �3
3 � sin �r �
2
2 � cos �
r �6
1 � cos �r �
5
1 � sin �
r �3
1 � sin �r �
2
1 � cos �
r �4
1 � 3 sin �r �
42 � cos �
r �2
1 � sin �r �
3
1 � 2 sin �
r �3
2 � cos �r �
2
1 � cos �
2
042
π2
042
π
2
02
π2
02 4
π
04
2π
2
02
π
r �4e
1 � e sin �r �
4e
1 � e sin �
r �4e
1 � e cos �r �
4e
1 � e cos �
e � 1.5.e � 0.5,e � 1,
Exercises 10.9
VOCABULARY CHECK:
In Exercises 1–3, fill in the blanks.
1. The locus of a point in the plane that moves so that its distance from a fixed point (focus) is in a constant ratio to its distance from a fixed line (directrix) is a ________.
2. The constant ratio is the ________ of the conic and is denoted by ________.
3. An equation of the form has a ________ directrix to the ________ of the pole.
4. Match the conic with its eccentricity.
(a) (b) (c)
(i) parabola (ii) hyperbola (iii) ellipse
PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.
e > 1e � 1e < 1
r �ep
1 � e cos �
333202_1009.qxd 12/8/05 9:09 AM Page 797
In Exercises 29–32, use a graphing utility to graph therotated conic.
29. (See Exercise 11.)
30. (See Exercise 16.)
31. (See Exercise 17.)
32. (See Exercise 20.)
In Exercises 33–48, find a polar equation of the conic withits focus at the pole.
Conic Eccentricity Directrix
33. Parabola
34. Parabola
35. Ellipse
36. Ellipse
37. Hyperbola
38. Hyperbola
Conic Vertex or Vertices
39. Parabola
40. Parabola
41. Parabola
42. Parabola
43. Ellipse
44. Ellipse
45. Ellipse
46. Hyperbola
47. Hyperbola
48. Hyperbola
49. Planetary Motion The planets travel in elliptical orbitswith the sun at one focus. Assume that the focus is at thepole, the major axis lies on the polar axis, and the length of the major axis is (see figure). Show that the polarequation of the orbit is where
is the eccentricity.
50. Planetary Motion Use the result of Exercise 49 to showthat the minimum distance ( ) from thesun to the planet is and the maximum dis-tance ( ) is
Planetary Motion In Exercises 51–56, use the results of Exercises 49 and 50 to find the polar equation of theplanet’s orbit and the perihelion and aphelion distances.
51. Earth
52. Saturn
53. Venus
54. Mercury
55. Mars
56. Jupiter
57. Astronomy The comet Encke has an elliptical orbit withan eccentricity of The length of the major axisof the orbit is approximately 4.42 astronomical units. Finda polar equation for the orbit. How close does the cometcome to the sun?
e � 0.847.
e � 0.0484a � 778.41 � 106 kilometers,
e � 0.0934a � 141.63 � 106 miles,
e � 0.2056a � 35.98 � 106 miles,
e � 0.0068a � 108.209 � 106 kilometers,
e � 0.0542a � 1.427 � 109 kilometers,
e � 0.0167a � 95.956 � 106 miles,
r � a�1 � e�.distanceaphelionr � a�1 � e�
distanceperihelion
2
0θ
r
a
Sun
Planet
π
er � a�1 � e2���1 � e cos ��
2a
�4, ��2�, �1, ��2��1, 3��2�, �9, 3��2��2, 0�, �8, 0��20, 0�, �4, ���2, ��2�, �4, 3��2��2, 0�, �10, ���10, ��2��5, ���6, 0��1, ���2�
x � �1e �32
x � 1e � 2
y � �3e �34
y � 1e �12
y � �2e � 1
x � �1e � 1
r �5
�1 � 2 cos�� � 2��3�
r �6
2 � sin�� � ��6�
r �3
3 � sin�� � ��3�
r �2
1 � cos�� � ��4�
798 Chapter 10 Topics in Analytic Geometry
58. Satellite Tracking A satellite in a 100-mile-highcircular orbit around Earth has a velocity ofapproximately 17,500 miles per hour. If this velocity ismultiplied by the satellite will have the minimumvelocity necessary to escape Earth’s gravity and it willfollow a parabolic path with the center of Earth as thefocus (see figure).
(a) Find a polar equation of the parabolic path of thesatellite (assume the radius of Earth is 4000 miles).
(b) Use a graphing utility to graph the equation youfound in part (a).
(c) Find the distance between the surface of the Earthand the satellite when
(d) Find the distance between the surface of Earth andthe satellite when � � 60�.
� � 30�.
0
2π
Parabolicpath
Circularorbit
4100miles
Not drawn to scale
�2,
Model It
333202_1009.qxd 12/8/05 9:09 AM Page 798
Section 10.9 Polar Equations of Conics 799
Synthesis
True or False? In Exercises 59–61, determine whether thestatement is true or false. Justify your answer.
59. For a given value of over the interval tothe graph of
is the same as the graph of
60. The graph of
has a horizontal directrix above the pole.
61. The conic represented by the following equation is anellipse.
62. Writing In your own words, define the term eccentricityand explain how it can be used to classify conics.
63. Show that the polar equation of the ellipse
is
64. Show that the polar equation of the hyperbola
is
In Exercises 65–70, use the results of Exercises 63 and 64 towrite the polar form of the equation of the conic.
65. 66.
67. 68.
69. Hyperbola One focus:
Vertices:
70. Ellipse One focus:
Vertices:
71. Exploration Consider the polar equation
(a) Identify the conic without graphing the equation.
(b) Without graphing the following polar equations,describe how each differs from the given polar equation.
(c) Use a graphing utility to verify your results in part (b).
72. Exploration The equation
is the equation of an ellipse with What happens tothe lengths of both the major axis and the minor axis whenthe value of remains fixed and the value of changes?Use an example to explain your reasoning.
Skills Review
In Exercises 73–78, solve the trigonometric equation.
73. 74.
75. 76.
77. 78.
In Exercises 79–82, find the exact value of the trigonometricfunction given that and are in Quadrant IV and
and
79. 80.
81. 82.
In Exercises 83 and 84, find the exact values of ,and using the double-angle formulas.
83.
84.
In Exercises 85–88, find a formula for for the arithmeticsequence.
85. 86.
87. 88.
In Exercises 89–92, evaluate the expression. Do not use acalculator.
89. 90.
91. 92. 29 P210P3
18C1612C9
a1 � 5, a4 � 9.5a3 � 27, a8 � 72
a1 � 13, d � 3a1 � 0, d � �14
an
3�
2< u < 2�tan u � ��3,
�
2< u < �sin u �
45
,
tan 2ucos 2u,sin 2u
sin�u � v�cos�u � v�sin�u � v�cos�u � v�
cos v � 1/�2.sin u � �35
vu
�2 sec � � 2 csc �
42 cot x � 5 cos
�
2
9 csc2 x � 10 � 212 sin2 � � 9
6 cos x � 2 � 14�3 tan � � 3 � 1
pe
e < 1.
r �ep
1 ± e sin �
r1 �4
1 � 0.4 cos �, r2 �
41 � 0.4 sin �
r �4
1 � 0.4 cos �.
�5, 0�, �5, ���4, 0��4, ��2�, �4, ���2��5, ��2�
x 2
36�
y 2
4� 1
x 2
9�
y 2
16� 1
x 2
25�
y 2
16� 1
x 2
169�
y 2
144� 1
r 2 ��b2
1 � e 2 cos2 �.
x 2
a 2�
y 2
b2� 1
r 2 �b2
1 � e 2 cos2 �.
x 2
a 2�
y 2
b2� 1
r 2 �16
9 � 4 cos� ��
4
r �4
�3 � 3 sin �
r �e��x�
1 � e cos �.
r �ex
1 � e cos �
� � 2�,� � 0e > 1
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